Date on Master's Thesis/Doctoral Dissertation

12-2021

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Powers, Robert

Committee Co-Chair (if applicable)

McMorris, Fred

Committee Member

McMorris, Fred

Committee Member

Riedel, Thomas

Committee Member

Kong, Maiying

Author's Keywords

Mathematical consensus; discrete mathematics

Abstract

Consensus functions on finite median semilattices and finite median graphs are studied from an axiomatic point of view. We start with a new axiomatic characterization of majority rule on a large class of median semilattices we call sufficient. A key axiom in this result is the restricted decisive neutrality condition. This condition is a restricted version of the more well-known axiom of decisive neutrality given in [4]. Our theorem is an extension of the main result given in [7]. Another main result is a complete characterization of the class of consensus on a finite median semilattice that satisfies the axioms of decisive neutrality, bi-idempotence, and symmetry. This result extends the work of Monjardet [9]. Moreover, by adding monotonicity as a fourth axiom, we are able to correct a mistake from the Monjardet paper. An attempt at extending the results on median semilattices to median graphs is given, based on a new axiom called split decisive neutrality. We are able to show that majority rule is the only consensus function defined on a path with three vertices that satisfies split decisive neutrality and symmetry.

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